Method of making an electrostatic actuator

ABSTRACT

A method of fabricating an electrostatic actuator with an intrinsic stress gradient is provided. An electrode is formed on a substrate and a support layer is formed over the electrode. A metal layer is deposited onto the support layer via a deposition process. Deposition process conditions are varied in order to induce a stress gradient into the metal layer. The intrinsic stress in the metal layer increases in the direction from the bottom to the top of the metal layer. The support layer under the electrode is removed to release the electrostatic actuator.

RELATED APPLICATION

This Application is a continuation of U.S. application Ser. No.09/944,867, filed Aug. 31, 2001, issued on Sep. 23, 2003 as U.S. Pat.No. 6,625,004, incorporated herein by reference as if set forth fullyherein

U.S. GOVERNMENT LICENSE RIGHTS

The U.S. Government has a paid-up license in this invention and theright in limited circumstances to require the patent owner to licenseothers on reasonable terms as provided for by the terms of DARPAcontract MDA972-00-C-0010.

TECHNICAL FIELD

The invention relates to electrostatic actuators, and more particularly,to electrostatic actuators with intrinsic stress gradient.

BACKGROUND OF THE INVENTION

Electrostatically actuated beams are one of the fundamental buildingblocks in Micro-Electro-Mechanical System (MEMS) devices, and findapplications in a variety of fields, such as communications, sensing,optics, micro-fluidics, and measurement of materials properties. In thefield of communications, electrostatically actuated MEMS variablecapacitors and RF switches are used in tunable RF filter circuits. Theelectrostatically actuated MEMS capacitors and RF switches offer severaladvantages over solid state varactor diodes, including a more linearresponse and a higher quality (Q) factor. For example, MEMS-tuned HighTemperature Superconductor (HTS) resonators employing electrostaticallyactuated capacitors have demonstrated a frequency tuning range of about7.5% with a Q factor above 2000 at a frequency of 850 MHz and atemperature of 77° K.

A conventional electrostatically actuated MEMS capacitor comprises afixed bottom electrode on a substrate, and a flexible cantilever beamthat acts as a top movable electrode of the capacitor. The cantileverbeam is secured at one end to the substrate by a rigid anchor. Thecapacitance of the electrostatically actuated capacitor is changed byapplying a bias voltage to the capacitor. The applied voltageestablishes an electrostatic force on the beam that deflects thecantilever beam, thereby changing the inter-electrode gap between thetop electrode (cantilever beam) and the bottom electrode, which in turnchanges the capacitance. Therefore, the change in the inter-electrodegap of the capacitor upon electrostatic actuation of the cantilever beamchanges the capacitance of the capacitor. The initial position of thecantilever beam with no applied bias voltage establishes the low (oroff) capacitance of the capacitor. A curved cantilever beam allows for alower capacitance in the off state (as compared to a straight one),which is desirable for many applications. A common approach for settingthe initial curvature (with no applied bias) of the cantilever beam isto construct the beam using a layered stack of two metals with differentCoefficients of Thermal Expansion (CTE). A beam constructed from twometals is often referred to as a bimetallic beam. The CTE mismatchbetween the two metals in the bimetallic beam produces a stress gradientin the beam that causes the beam to curve as the temperature is changed.In order to achieve an initial upward curvature of the bimetallic beamat cryo-temperatures, the metal with the higher CTE (e.g., aluminum) isdeposited on top of the metal with the lower CTE (e.g., gold).

Although the usefulness of conventional electrostatically actuateddevices has been demonstrated, there are a number of problems with thesedevices stemming from the use of the bimetallic beam. The mechanicalproperties of the bimetallic beam have been shown to change with timedue to inter-metallic diffusion between the two metals andrecrystallization, thereby limiting the long term reliability andreproducibility of these devices. The introduction of a diffusionbarrier layer between the two metal layers can help reduceinter-metallic diffusion but can not solve it completely. Furthermore,the operation of these devices requires precise temperature controlbecause any variation in temperature leads to a change in the deflectionof the bimetallic beam, which in turn changes the capacitance of thesedevices.

Ideally the two metals used to construct the bimetallic beam areductile, resistant to fatigue or work hardening, electrical conductive,and easy to process. However, the requirement that the two metals havedifferent CTEs in order to achieve a desired initial curvature restrictsthe types of metal that are available to construct the bimetallic beam.Typically, a less than ideal metal (e.g., indium, aluminum or zinc) hasto be used for one of the two metals of the bimetallic beam in order toachieve a desired initial curvature. As a result, the mechanical andelectrical properties of the cantilever beam are compromised.

SUMMARY OF THE INVENTION

The present invention provides an electrostatic actuator with intrinsicstress gradient. The electrostatic actuator according to the inventionuses a metal layer made substantially of a single metal with anintrinsic stress gradient induced therein instead of a bimetallic beam.

An electrostatic actuator, built in accordance with an embodiment of theinvention, comprises an electrode and an electrostatically actuated beamfixed at one end relative to the electrode. The electrostaticallyactuated beam comprises a metal layer made substantially of a singlemetal with a stress gradient induced therein. The stress gradient in themetal layer causes the beam to curve away from the electrode. Uponelectrostatic actuation of the beam, the beam is deflected from itsinitial position relative to the electrode.

An electrostatically actuated variable capacitor, built in accordancewith another embodiment of the invention, comprises a substrate, abottom electrode on the substrate, a curved cantilever beam acting as atop movable electrode of the capacitor, and a rigid anchor for securingone end of the cantilever beam to the substrate. The cantilever beamcomprises a metal layer made substantially of a single metal with astress gradient induced therein. The stress gradient in the metal layercauses the beam to initially curve upwardly from the substrate. Uponelectrostatic actuation of the beam, the beam is deflected from itsinitial curvature relative to the bottom electrode, thereby changing theintro-electrode gap between the top electrode (cantilever beam) and thebottom electrode and the capacitance of the capacitor.

In one embodiment of the invention, the stress gradient is induced inthe metal layer of the beam by varying process conditions duringdeposition of the metal layer.

In another embodiment of the invention, the metal layer is depositedusing electroplating. In this embodiment, the stress gradient is inducedin the metal layer of the beam by varying the electroplating bath,current density, and/or temperature during electroplate deposition ofthe metal layer.

In another embodiment, the metal layer is deposited in a two stepdeposition process of a soft metal deposition followed by a hard metaldeposition, where the deposited hard metal exhibits higher intrinsicstress than the soft metal. The soft and hard metal are both made of thesame metal, e.g., gold, characterized by different crystal grain sizes.In one embodiment, the electroplating bath used to deposit the hardmetal contains dopants to enhance the intrinsic stress in the hardmetal.

In another embodiment of the invention, the metal layer is depositedusing evaporation, such as electron beam evaporation. In thisembodiment, the stress gradient is induced in the metal layer by varyingthe deposition rate during deposition of the metal layer. In oneembodiment, an initial upward curvature of the beam is obtained byincreasing the deposition rate during deposition of the metal layer.

In another embodiment of the invention, the metal layer is depositedusing sputtering. In this embodiment, the stress gradient is induced inthe metal layer by varying the gas pressure of the sputtering duringdeposition of the metal layer. In one embodiment, an initial upwardcurvature of the beam is obtained by increasing the gas pressure of thesputtering during deposition of the metal layer.

In another embodiment of the invention, the cantilever beam is fullyhinged to the anchor at one end.

In another embodiment of the invention, the cantilever beam is doublehinged to the anchor at one end.

In another embodiment of the invention, the front edge of the cantileverbeam is straight.

In another embodiment of the invention, the front edge of the beam iscurved to alleviate stress concentration due to sharp corners.

Other objects and features of the present invention will become apparentfrom consideration of the following description taken in conjunctionwith the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional view of an electrostatically actuatedcapacitor according to an embodiment of the invention.

FIG. 2 is a process sequence for fabricating an electrostaticallyactuated capacitor according to an embodiment of the invention.

FIG. 3 is a process sequence for fabricating an electrostaticallyactuated capacitor according to another embodiment of the invention.

FIG. 4 is a graph of the induced bending moment of a cantilever beam asa function of hard gold thickness.

FIG. 5 is a graph of the stress gradient of a cantilever beam as afunction of hard gold thickness

FIG. 6 is a top view of five cantilever beams built in accordance withthe invention.

FIG. 7 is a graph of the deflection-voltage behavior of a double-hingedsquare beam according to an embodiment of the invention.

FIG. 8 is a graph of the deflection-voltage behavior of a full-hingedsemicircle beam according to an embodiment of the invention.

FIG. 9 is a graph of the deflection-voltage behavior of a double-hingedelliptical beam according to an embodiment of the invention.

FIG. 10 is a graph of the deflection-voltage behavior of a full-hingedsquare beam according to an embodiment of the invention.

FIG. 11 is a graph of the capacitance of the full-hinged square beam asa function of applied bias voltage.

FIG. 12 is a graph of the deflection-voltage behavior of a double-hingedrectangle beam according to an embodiment of the invention.

FIG. 13 is a top view of a cantilever beam according to an embodiment ofthe invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is believed to be applicable to a variety ofelectrostatic actuators requiring electrostatically actuated beams. Thepresent invention is particularly applicable to electrostaticallyactuated variable capacitors. While the present invention is not solimited, an appreciation of the present invention is best presented byway of a particular example application, in this instance, in thecontext of such electrostatically actuated variable capacitors.

Turning to the drawings, FIG. 1 shows a cross section of anelectrostatically actuated capacitor 10 according to an embodiment ofthe invention. The capacitor 10 comprises a base substrate 30, a bottomelectrode 20 on the substrate 30 and a curved cantilever beam 15, whichact as a top movable electrode of the capacitor 10. The cantilever beam15 according to the invention comprises a metal layer with an intrinsicstress gradient induced therein. The metal layer is substantially madeof a single metal. The stress gradient in the metal layer produces aninitial upward curvature in the cantilever beam 15. The stress gradientis induced in the metal layer of the beam 15 by varying processconditions during deposition of the metal layer. A more detaileddiscussion will follow. The capacitor 10 further comprises a rigidanchor 25 for securing one end of the cantilever beam 15 to thesubstrate 30. The rigid anchor 25 may be made of the same material asthe beam 15. The capacitor 10 also includes an optional dielectric layer27 deposited over the bottom electrode 20. Preferably, the metal layerof the cantilever beam 15 is made of a metal characterized by highelectrical conductivity, low loss, ease of deposition, and excellentflexibility. Suitable metals for the metal layer include gold andsilver. The bottom electrode 20 may be made of aluminum on, e.g., asilicon substrate. Alternatively, the bottom electrode 20 may be made ofa thin-film High Temperature Superconductor (HTS) material on, e.g., aMgO substrate. Thin-film HTS materials are now routinely formed and arecommercially available. See, e.g., U.S. Pat. Nos. 5,476,836, 5,508,255,5,843,870, and 5,883,050. Also see, e.g., B. Roas, L. Schultz, and G.Endres, “Epitaxial growth of YBa2Cu3O7-x thin films by a laserevaporation process” Appl. Phys. Lett. 53, 1557 (1988) and H. Maeda, Y.Tanaka, M. Fukotomi, and T. Asano, “A New High-Tc Oxide Superconductorwithout a Rare Earth Element” Jpn. J. Appl. Phys. 27, L209 (1988).

The electrostatically actuated capacitor 10 according to the inventionmay be fabricated by first forming the bottom electrode 20 and theanchor 25 on the substrate 30 using well-known metal deposition andetching techniques. A release layer (not shown), e.g., photoresist, isdeposited over the bottom electrode 20 to provide mechanical support forthe cantilever beam 25 during fabrication. A metal layer is thendeposited onto the release layer and the anchor 25 to form thecantilever beam 15. After the cantilever beam 25 is formed, the releaselayer underlying the cantilever beam 25 is removed to release the beam25. For a release layer made of photoresist, acetone may be used toremove the support layer.

To induce a stress gradient in the metal layer of the beam 25, theintrinsic stress of the deposited metal forming the metal layer isincreased during deposition of the metal layer. This way, the intrinsicstress in the metal layer increases in the direction from the bottom tothe top of the metal layer. This results in an intrinsic stress gradientin the metal layer that causes the cantilever beam 25 to curve upwardlyas shown in FIG. 1. The intrinsic stress of the deposited metal can beincreased during deposition of the metal layer by increasing thedeposition rate and/or lowering the deposition temperature. This isbecause increasing the deposition rate and/or lowering the depositiontemperature tends to reduce the ability of the deposited metal atoms toform complete bonds with each other, resulting in smaller crystal grainsizes in the deposited metal. The smaller crystal grain sizes in turnincrease the intrinsic stress in the deposited metal. Therefore, astress gradient can be induced in the metal layer of the beam 25 byincreasing the deposition rate and/or lowering the depositiontemperature during deposition of the metal layer. The deposition ratecan be increase and/or the deposition temperature can be loweredcontinuously or incrementally during deposition of the metal layer.

The intrinsic stress in the deposited metal can also be increased duringdeposition by doping the deposited metal with dopants, e.g., cobalt,which induce stress in the deposited metal. The higher the doping level,i.e., concentration, in the deposited metal, the greater the intrinsicstress induced in the deposited metal. Therefore, a stress gradient canbe induced in the metal layer by increasing the doping level in thedeposited metal during deposition of the metal layer. The doping levelin the deposited metal can be increased in addition to increasing thedeposition rate and/or the lowering the deposition temperature toenhance the stress gradient in the metal layer.

The present invention contemplates at least three deposition techniquesthat can be used to induce the stress gradient in the metal layernamely, electroplating, evaporation, and sputtering. Usingelectroplating deposition, the intrinsic stress in the metal layer canbe increased during electroplate deposition by varying theelectroplating bath composition, current density and/or temperature. Inone embodiment, a two step electroplate deposition process of a softmetal electroplate deposition followed by a hard metal electroplatedeposition is used to form the metal layer of the cantilever beam 15.The deposited soft metal and the hard metal are both made of the samemetal characterized by different crystal grain sizes. More particularly,the deposited hard metal is characterized by smaller crystal grainsizes, and therefore higher intrinsic stress, than the deposited softmetal. The hard metal can be realized by depositing the hard metal at ahigher current density and/or lower temperature than the soft metal,where the higher current density translates into a higher depositionrate. The hard metal can also be realized by doping the hard metal at ahigher doping level than the soft metal so that the hard metal hashigher stress than the soft metal.

Using evaporation, the intrinsic stress in the metal layer can beincreased during deposition by increasing the deposition rate of themetal layer, either continuously or incrementally. For example, anelectron beam evaporator may be used to vary the gold deposition rate,e.g., between 0.1 and 1 nm/sec, during deposition of the metal layer inorder to induce an intrinsic stress gradient in the metal layer.

Using sputtering (e.g., DC sputtering at 300 W power), the intrinsicstress in the metal layer can be increased during deposition byincreasing the gas pressure of the sputtering, either continuously orincrementally. For example, the gas pressure of the sputtering may bevaried between 0.8 and 2.4 Pa during deposition of the metal layer inorder to induce an intrinsic stress gradient in the metal layer.

The deposition techniques discussed so far induce a stress gradient inthe metal layer, in which the intrinsic stress increases along thethickness of the metal layer. The stress gradient can also be made toincrease along the length of the metal layer. An example of one way ofdoing this is shown in FIG. 13, which shows a top view of a rectangularcantilever beam 1310 having a length, L, and a width, W, according to anembodiment of the present invention. The metal layer of the beam 1310comprises deposited soft metal 1320 and deposited hard metal 1330 on topof the soft metal 1320. The deposited soft metal 1320 and the hard metal1330 are both made of the same metal, i.e., gold, characterized bydifferent crystal grain sizes. More particularly, the deposited hardmetal 1330 is characterized by smaller crystal grain sizes, andtherefore higher intrinsic stress, than the deposited soft metal 1320.The deposited soft gold 1320 is patterned to be rectangular in shape.The deposited hard metal, however, is patterned to be triangular inshape such that the width of the hard metal 1330 increases from thehinged, i.e., anchored, end 1340 of the beam 1310 to the tip end 1350 ofthe beam 1310. This produces a stress gradient in the beam 1310, inwhich the intrinsic stress increases along the length of the beam 1310in moving from the hinged end 1340 of the beam 1310 to the tip end 1350of the beam 1310. Even though the hard layer 1330 was triangular inshape in this particular example, those skilled in the art willappreciate that other shapes which increase the width of the hard layer1330 from the hinged end 1340 to the tip end 1350 may also be used.

In operation, the initial curvature of the cantilever beam 15 producedby the stress gradient of the metal layer determines the low (or off)capacitance of the capacitor 10 with no applied bias voltage. The beamis electrostatically actuated by applying a bias voltage to thecapacitor 10. The bias voltage establishes an electrostatic force on thecantilever beam 15 that deflects the beam 15 from its initial curvaturerelative to the bottom electrode 20. This beam deflection changes theinter-electrode gap 35 between the beam 15 and the bottom electrode 20,thereby changing the capacitance of the capacitor 10. Therefore, thecapacitance of the electrostatically actuated variable capacitor 10 canbe tuned by varying the bias voltage applied to the capacitor 10. Theelectrostatically actuated variable capacitor 10 according to theinvention may be used in a variety of applications requiring highlyselective tunable filters with a high Q factor, for example, byincorporating the electrostatically actuated variable capacitor into afilter circuit.

The cantilever beam 15 according to the invention offers severaladvantages over the bimetallic cantilever beam. One advantage is thatthe curvature of the cantilever beam 15 is relatively independent oftime and temperature. This is because the curvature of the cantileverbeam 15 is produced by an intrinsic stress gradient in a metal layermade of a single metal instead of a CTE mismatch between two differentmetals. Because the curvature of the beam 15 is relatively independentof temperature, the capacitance of the capacitor 10 does not vary withtemperature. As a result, the capacitor 10 according to the presentinvention does not require precise temperature control for itsoperation. In addition, the mechanical and electrical properties of thecantilever beam 15 are not compromised by having to select two differentmetals with different CTEs to set the initial curvature of the beambecause a single metal type is used. Furthermore, problems between thetwo different metal layers of the bimetallic beam, such asinter-metallic diffusion, are eliminated.

Those skilled in the art will appreciate that cantilever beam 15according to the invention is not limited to use in a variablecapacitor. For example, the cantilever beam 15 according to theinvention can be used in an electrostatically actuated RF switch. An RFswitch is typically similar in construction to the variable capacitorwith no dielectric layer deposited over the bottom electrode so that thetip of the cantilever beam can make contact with the bottom electrode.When no bias voltage is applied to the beam, the beam does not contactthe bottom electrode and the RF switch is open. When the beam iselectrostatically actuated by a sufficient bias voltage, the beam ispulled down to make contact with the bottom electrode, thereby closingthe RF switch.

Two analytical models of the cantilever beam according to the inventionwill now be developed. The first analytical model is used to model theinitial curvature, i.e., deflection, of the cantilever beam withintrinsic stress gradient. In this model, a cantilever beam withdimensions of length, L, width, w, and thickness, b, is used in thederivation of the initial deflection of the beam. The beam is assumed tobe fully clamped at one end and free at the other end. The stress in thetop and bottom portion of the beam is assumed to be σ₁ and σ₂respectively. Further, it is assumed that the in-plane deformation ofthe beam is negligible compared to its out-of-plane deformation.Assuming a linear stress profile within the beam, the stress can bedivided into two components, as given in Equation 1, $\begin{matrix}{{Stress} = {\left( \frac{\sigma_{1} + \sigma_{2}}{2} \right) + {\left( \frac{\sigma_{1} - \sigma_{2}}{b} \right)(y)}}} & \left( {{Eq}.\quad 1} \right)\end{matrix}$where y is taken along the thickness of the beam. The uniform stress,$\left( \frac{\sigma_{1} + \sigma_{2}}{2} \right)$accounts for the in-plane elongation of the beam whereas, the$\left( \frac{\sigma_{1} - \sigma_{2}}{b} \right)(y)$(the stress gradient component) causes out of plane deflection. The netforce on the element of area wdy is given by Equation 2, $\begin{matrix}{{\mathbb{d}F} = {\left( \frac{\sigma_{1} - \sigma_{2}}{b} \right){y \cdot w \cdot {\mathbb{d}y}}}} & \left( {{Eq}.\quad 2} \right)\end{matrix}$The moment about the axis through the center of the plane is given byEquation 3, $\begin{matrix}{{\mathbb{d}M} = {{{\mathbb{d}F} \cdot y} = {\left( \frac{\sigma_{1} - \sigma_{2}}{b} \right) \cdot y \cdot w \cdot {\mathbb{d}y} \cdot y}}} & \left( {{Eq}.\quad 3} \right)\end{matrix}$The net moment acting on the beam cross section can be obtained byintegrating the Equation 3 over the thickness of the beam, as shown byEquation 4, $\begin{matrix}{{\int_{{- b}/2}^{b/2}\quad{\mathbb{d}M}} = {\int_{{- b}/2}^{b/2}\quad{\left( \frac{\sigma_{1} - \sigma_{2}}{b} \right) \cdot y^{2} \cdot w \cdot {\mathbb{d}y}}}} & \left( {{Eq}.\quad 4} \right)\end{matrix}$Using symmetry within the beam cross section, the net moment acting onthe beam, M is given by Equation 5, $\begin{matrix}{M = {2 \cdot \left( \frac{\sigma_{1} - \sigma_{2}}{b} \right) \cdot \frac{b^{3}}{8 \cdot 3} \cdot w}} & \left( {{Eq}.\quad 5} \right)\end{matrix}$Therefore, the stress induced bending moment in the beam can beexpressed by Equation 6, $\begin{matrix}{M = {\left( \frac{\sigma_{1} - \sigma_{2}}{12} \right) \cdot b^{2} \cdot w}} & \left( {{Eq}.\quad 6} \right)\end{matrix}$For a cantilever beam fixed at one end, with moment M acting on theother end, the governing equation for static deflection from continuummechanics is given by Equation 7, $\begin{matrix}{{{EI}\frac{\mathbb{d}^{2}z}{\mathbb{d}x^{2}}} = M} & \left( {{Eq}.\quad 7} \right)\end{matrix}$where, z is the deflection of the cantilever beam, E is elastic modulusof the material and I is the moment of inertia (wb³/12). For the beamfixed at one end, the boundary conditions used are x=0, z=0, and x=0,(dz/dx)=0. Solving the differential equation, Equation. 7 with the aboveboundary conditions, and solving for the deflection, z, gives Equation8, $\begin{matrix}{z = {{\frac{M}{EI}\frac{x^{2}}{2}} = {{\left( \frac{\sigma_{1} - \sigma_{2}}{12} \right) \cdot \frac{b^{2}w}{EI} \cdot \frac{x^{2}}{2}} = {\left( \frac{\sigma_{1} - \sigma_{2}}{12} \right) \cdot \frac{b^{2}w}{EI} \cdot \frac{x^{2}}{2}}}}} & \left( {{Eq}.\quad 8} \right)\end{matrix}$where z gives the height of the beam with respect to the fixed end (x=0)of the beam as a function of position, x, along the length of the beam.

The second analytical model is used to mathematically model thedeflection of the cantilever beam as a function of applied voltage. Inthis model, the initial curvature of the cantilever beam is modeled asif the beam were made of two different materials. This is done strictlyto provide a mathematically model of the cantilever beam, and is notintended to describe the actual physical composition of the beam. See,e.g., S. Timoshenko, “Analysis of bi-metal thermostats”, Journal ofstructural analysis & R.S.I, September, 1925, and Y. H. Min, Y. K. Kim,“In situ measurement of residual stress in micromachined thin filmsusing a specimen with composite-layered cantilevers”, J. Micromech,Microeng. 10 (2000) 314-321. In this model, the stress gradient of thebeam can be created by using two materials with dissimilar coefficientof thermal expansion (CTE). Due to the CTE mismatch between the twomaterials, the cantilever beam can be made to deflect away from itsoriginal position in the positive (upward) or in the negative (downward)direction by changing the temperature. The theory for modeling the beamusing this approach is presented as follows.

The change in radius of curvature of a bimetallic cantilever beam wouldbe same as that of a free standing bimetallic strip of twice the lengthof the cantilever beam. Assuming that the bimetallic beam is planar(flat) at room temperature, the final radius of curvature of thefreestanding film can be calculated. If there are no external forcesacting on the bimetallic beam, thenP ₁ =P ₂ =P(equilibrium)  (Eq. 9)$\begin{matrix}{{P\left( \frac{t_{1} + t_{2}}{2} \right)} = {M_{1} + M_{2}}} & \left( {{Eq}.\quad 10} \right)\end{matrix}$where M₁ and M₂ are the moments given by${M_{1} = \frac{E_{1}I_{1}}{\rho}},{M_{2} = \frac{E_{2}I_{2}}{\rho}},$P₁, P₂ are the external forces applied on the top and bottom layer,respectively, of the beam, and ρ is the radius of curvature of the beam.The moment of inertia of the top layer, I₁, is given by (b*t₁ ³/12) andthe moment of inertia of the bottom layer, I₂, is given by (b*t₂ ³/12),where, b is the width of the cantilever beam, and t₁ and t₂ are thethickness of the top and bottom layer respectively. Also, E₁, E₂ are theElastic modulus of the materials of the top and bottom layer,respectively, of the beam. Due to the CTE mismatch, the strain developedin the two materials must be equal which leads to Equation 11,$\begin{matrix}{{{\alpha_{1}\left( {\Delta\quad T} \right)} + \frac{P}{E_{1}\left( {bt}_{1} \right)} + \frac{t_{1}}{2\rho}} = {{\alpha_{2}\left( {\Delta\quad T} \right)} - \frac{P}{E_{2}\left( {bt}_{2} \right)} - \frac{t_{2}}{2\rho}}} & \left( {{Eq}.\quad 11} \right)\end{matrix}$where, α₁ and α₂ are the CTE's of material of two layers, and ΔT is thechange in temperature, Substituting the expressions for P from Equation10, and for the beam with initial radius of curvature of ρ_(o), thechange in radius of curvature is given by Equation 12, $\begin{matrix}{{\frac{1}{\rho} - \frac{1}{\rho_{o}}} = \frac{\left( {\alpha_{2} - \alpha_{1}} \right)\left( {\Delta\quad T} \right)}{\frac{h}{2} + {\frac{2\left( {{E_{1}I_{1}} + {E_{2}I_{2}}} \right)}{bh}\left( {\frac{1}{E_{1}t_{1}} + \frac{1}{E_{2}t_{2}}} \right)}}} & \left( {{Eq}.\quad 12} \right)\end{matrix}$where h=t₁+t₂. The initial deflection of the bimetallic cantilever beamalong the length, z(x) is given by Equation 13, $\begin{matrix}{{z(x)} = {\frac{x^{2}}{2}\left( {\frac{1}{\rho} - \frac{1}{\rho_{o}}} \right)}} & \left( {{Eq}.\quad 13} \right)\end{matrix}$Thus, the initial deflection of the curved actuator can be modeled ifthe thickness, elastic modulus, and CTE of the materials are known.

The electrostatic force acting on the movable electrode is given by${F = \frac{ɛ\quad A\quad V^{2}}{2g^{2}}},$where, ε is the permittivity of the medium, A is the area of theelectrodes, V is the applied voltage, g is the gap distance between thebeam and the bottom electrode, which is equal to the sum of the initialgap distance, z, and displacement, y, between the fixed end of the beamand the bottom electrode. The force per unit length acting on thecantilever beam is $\frac{ɛ\quad V^{2}}{2g^{2}} \times {b.}$From the continuum mechanics, $\begin{matrix}{{{EI}\frac{\mathbb{d}^{4}y}{\mathbb{d}x^{4}}} = {- \frac{V^{2}ɛ\quad b}{2\left( {y + z} \right)^{2}}}} & \left( {{Eq}.\quad 14} \right)\end{matrix}$where, EI is the flexural rigidity of the beam, and z is the height ofthe beam as a function of distance, x, along the length of the beam fromthe fixed end of the beam, x=0. The height z(x) can be obtained eitherby using the stress gradient model of the first model or byapproximating the beam by a bimetal composite and matching the initialdeflection. Equation 14 is a fourth order nonlinear differentialequation, which can be solved for z(x) using following boundaryconditions.

-   1. y(x=0)=0 No deflection at the fixed end of the beam.-   2. ${\frac{\mathbb{d}y}{\mathbb{d}x}\left( {x = 0} \right)} = 0$-    Slope of the deflection at the fixed end is zero.-   3.    ${\frac{\mathbb{d}^{2}y}{\mathbb{d}x^{2}}\left( {x = L} \right)} = 0$-    No moment acting on the free end.-   4.    ${\frac{\mathbb{d}^{3}y}{\mathbb{d}x^{3}}\left( {x = L} \right)} = 0$-    No shear force acting on the free end.    Using the above boundary conditions, Equation 14 can be solved    numerically for different points along the beam by employing finite    difference techniques. To model the deflection of an actual    cantilever beam, built in accordance with the invention, the initial    curvature of the model beam is matched to the measured initial    curvature of the actual beam. This can be done, example, by    adjusting the values of the CTE values, α₁ and α₂, of the two    materials of the model beam until the initial curvature of the model    beam matches the measured curvature of the actual beam. Once the    initial curvature of the model beam is matched to the curvature of    the actual beam, the model beam can be used to model the deflection    of the actual beam as a function of voltage by varying the applied    voltage, V, of the model beam.

Once the height, z(x), of the cantilever beam is known as a function ofposition, x, along the length of the beam, the capacitance between thebeam (top electrode) and the bottom electrode can be calculated usingEquation 15, $\begin{matrix}{C = {\int_{0}^{L}\frac{ɛ_{o}w{\mathbb{d}x}}{\left( {g_{air} + {\frac{1}{k}g_{dielectric}}} \right)}}} & \left( {{Eq}.\quad 15} \right)\end{matrix}$where g_(air) is the gap between the beam and the bottom electrode withair as the dielectric medium, ε is the permittivity of air, k is therelative permittivity of a dielectric layer deposited over the bottomelectrode, and g_(dielectric) is the thickness of the dielectric layer.The gap, g_(air), between the beam and the bottom electrode with air asthe dielectric medium can be determined by adding the height z(x) of thebeam with respect to the height of the fixed end (x=0) of the beam, tothe height of the fixed end of the beam with respect to the dielectriclayer. The height z(x) of the beam can be determined using the first orsecond model. Equation 15 can be numerically integrated along thelength, L, of the beam to determine the capacitance of the device.

An exemplary sequence of process steps for fabricating anelectrostatically actuated capacitor according to an embodiment of theinvention will now be described with respect FIG. 2. In step (a) of FIG.2, a silicon wafer (e.g., (100) n-SI wafer) with a thermally grown oxidelayer (e.g., 400 nm thick) is used as a base substrate 120 for thecapacitor. In addition, an aluminum layer 130 (e.g., 0.4 μm thick) isdeposited onto the substrate using a DC magnetron sputterer. In step(b), the aluminum layer 130 is patterned to form the bottom electrode135 and a base for the anchor 132 of the capacitor. In step (c), a 1.6μm thick dielectric polymer 145 is deposited onto the wafer andpatterned using a photoresist (Shipley Chemical Co. 1800 series) to forma via for contact to the top electrode (cantilever beam). In step (d), atitanium layer (e.g., 30 nm thick) followed by a gold layer (e.g., 1 μmthick) are deposited onto the wafer using electron-beam evaporation (CVCproducts) at a rate of 0.3 nm/sec. In step (e), the gold and titaniumlayers are patterned to form the cantilever beam anchor 147. A potassiumiodide etchant (potassium iodide 100 g/l and iodine 25 g/l) may be usedto etch the gold layer. An EDTA etchant (0.1 Methylenediaminetetraacetate, disodium salt dihydrate solution, adjustedto a pH of 11 with 5% H₂O₂) may be used to etch the titanium layer. Instep (f), a Shipley 1800 series photoresist is spun onto the wafer at3000 rpm to a thickness of 1.5 μm and patterned to form a photoresistrelease layer 150 for the cantilever beam. In step (g), a titanium andgold layer is deposited onto the wafer and patterned to form thecantilever beam 155. In this embodiment, the gold layer forms the metallayer with the intrinsic stress gradient.

A two step deposition process of a soft gold electroplate depositionfollowed by a hard gold electroplate deposition is used to form the goldlayer of the cantilever beam 155. A titanium and gold layer (20 nm and200 nm thick, respectively) deposited at a rate of 0.2 nm/sec onto usinge-beam evaporation is used as a seed layer for the electroplatedeposition of the soft and hard gold. The deposited soft gold ischaracterized by low residual intrinsic stress. A buffered cyanide goldplating bath (KAu(CN)₂ 20 g/l, K₂HPO₄ 40 g/l, KH₂PO₄ 10 g/l adjusted toa pH of 7) at a current density of 2 to 10 mA/cm² at a temperature of60° C. may be used to deposit the soft gold and an acid cyanide platingbath (KAu(CN)₂ 15 g/l, citric acid 50 g/l, cobalt (added as acetate)0.07 g/l, adjusted to a pH of 3.5) at a current density of 5 MA/cm² androom temperature may then be used to deposit the hard gold. The acidcyanide plating bath dopes the hard gold with cobalt to further enhancethe intrinsic stress of the hard gold. Even though different goldplating baths were used to deposit the soft and hard gold, the same goldplating bath could have been used to deposit both the soft and hardgold. In addition, even though gold was deposited, the invention can beextended to other types of metals, which can be deposited usingelectroplating, such as silver.

In step (h), the photoresist release layer 150 is removed using acetoneto release the cantilever beam 155. The titanium adhesion layer belowthe gold layer of the cantilever beam 155 is etched using the EDTAsolution. The released cantilever beam 155 is then rinsed usingisopropanol and methanol, and dried in the nitrogen-purged oven at 90°C.

If the cantilever beam is immediately dried after it is rinsed inmethanol, the cantilever beam 155 may experience stiction due to thesurface tension of the methanol. The surface tension of the drainingliquid (methanol) may draw the cantilever beam 155 into contact with theunderlying substrate. A Self-Assembled Monolayer (SAM) may be used tocreate a hydrophobic surface on the cantilever beam 155 to eliminate thein-use stiction problem. The SAM may be formed by soaking the releasedcantilever beam 155 in a 1 mM dodecyl-thiol solution.

A sequence of process steps for fabricating an electrostaticallyactuated capacitor according to another embodiment is shown in FIG. 3.The capacitor according to this embodiment has a segmented bottomelectrode. In step (a) of FIG. 3, an aluminum layer 210 (e.g., 400 nmthick) is deposited onto a base substrate 220 using DC magnetronsputterer. In step (b) the aluminum layer 210 is patterned to form asegmented bottom electrode 232 and a base 235 for the anchor. A polymermaterial (e.g. 1.6 μtm thick) is also deposited to form a dielectriclayer 240 over the bottom electrode 232. In step (c), the dielectriclayer 240 is patterned to form a via 245 for the anchor. In step (d), aphotoresist layer 250 (e.g., 1.5 μm thick) is deposited onto the waferand patterned to expose the via 245. In step (e), a titanium (e.g., 30mm thick) and gold (e.g., 1 μm thick) layer 255 is deposited onto thewafer using electron beam evaporation (CVC products). In step (f), thetitanium and gold layer 255 is patterned to form the anchor 255 a in via245. In step (g), a photoresist layer is deposited (e.g., 30 nm thick)and removed from the via 245 to form a release photoresist layer 265.Titanium (e.g. 30 nm thick) followed by gold (e.g., 200 nm thick) aredeposited on top of the photo release layer 265 and anchor 255 a to forma seed layer for electroplate deposition. In step (h), a soft gold layer270 is electroplate deposited using a buffered cyanide gold plating bath(KAu(CN)₂ 20 g/l, K₂HTPO₄ 40 g/l, KH₂PO₄ 10 g/l adjusted to a pH of 7)at current density of 3 mA/cm² and a temperature of 60° C. In step (i),a hard gold layer 275 is electroplate deposited onto the soft gold layer270 in an acid cyanide plating bath (KAu(CN)₂ 15 g/l, citric acid 50g/l, cobalt (added as acetate) 0.07 g/l, adjusted to a pH of 3.5) at acurrent density of 5 mA/cm² and room temperature. The hard gold layer275 exhibits higher intrinsic stress than the underlying soft gold layer270 due to the smaller crystal grain size of the hard gold layer 275. Instep (j), the soft gold layer 270 and the hard gold layer 1275 arepatterned, e.g., using conventional lithographic techniques, to form acantilever beam 280 of the capacitor. The cantilever beam 280 isreleased by dissolving the photoresist release layer 265 in acetone.Stiction during the release of the cantilever beam 280 may be reduced bytreating the cantilever beam with an alkane thiol solution to form ahydrophobic Self-Assembled Monolayer (SAM) on the gold. The SAM may beformed by soaking the released cantilever beam 280 in a 1 mM solution ofdodecyl thiol in an absolute ethanol solution.

Four actuators with full-hinged cantilever beams were built inaccordance with the present invention using the process in FIG. 2. Eachone of the cantilever beams had a length and width of 1 mm and 0.5 mm,respectively. In addition, each one of the cantilever beams had a softgold thickness of about 2.2 μm and a different hard gold thickness. Thehard gold thicknesses for the four cantilever beams were 0, 0.1, 0.2 and0.3 μm.

The tip angle for each one of the four cantilever beams was measuredusing Scanning Electron Microscopy (SEM) imaging. The tip angle of eachcantilever beam gave a direct measure of the deflection of thecantilever beam. The cantilever beam with the zero hard gold thicknessexhibited no curvature. This indicates that the residual stress in thesoft gold layer alone was not sufficient to overcome the van der Waalsand surface forces acting on the cantilever beam. The cantilever beamwith the hard gold thickness of 0.1 μm had a tip angle of 18° and aradius of curvature of 3.2 mm. The cantilever beam with the hard goldthickness of 0.2 μm had a tip angle of 350 and a radius of curvaturedecreased to 1.64 mm. Finally, the cantilever beam with the hard goldthickness of 0.3 μm had a tip angle of 72° and a radius of curvature of795 μm. From the above tip angle values, it can be seen that the initialdeflection of the cantilever beam is directly proportional to thethickness of the hard gold.

FIG. 4 is a graph of the estimated stress induced bending moments (Nm)for the above cantilever beams plotted as a function of hard goldthickness. The induced bending moment gives a measure of the amount ofout-of-plane force exerted on a cantilever beam due to the deposition ofthe hard gold. The estimated stress induced bending moments plotted inFIG. 4 are calculated using Equation 6,$M = {\left( \frac{\sigma_{1} - \sigma_{2}}{12} \right) \cdot b^{2} \cdot w}$where σ₁ is the stress of the hard gold layer, σ₂ is the stress of thesoft gold layer (assumed to be zero), b is the thickness of the beam,and w is the width of the beam. The stress of the hard gold σ₁ isestimated from the curvature of the beam. The bending moment for thebeam deposited with 0.1 μm of hard gold is 86×10⁻¹⁰ Nm. The estimatedbending moments for the hard gold thickness of 0.1, 0.2 and 0.3 μm werefitted to a line using a least-squares fit method. The degree of fit ofthe estimated bending moments to the line is R²=0.978, where R²=1indicates a perfect linear fit. It can be seen from FIG. 4 that thebending moment acting on the cantilever beam is approximately linearlyproportional to hard gold thickness.

FIG. 5 is a graph of calculated values of the intrinsic stress gradientin the cantilever beams plotted as a function of hard gold thickness.The residual stress gradients plotted in FIG. 5 are calculated usingEquation 8, where a bulk elastic modulus, E, of 74 GPa is used for thegold. The effective residual stress gradient in the cantilever beamdeposited with 0.1 μm of hard gold is 15.5 MPa/μm. The calculated stressgradients for the hard gold thickness of 0.1, 0.2 and 0.3 μm are fittedto a line using a least-squares fit method. The degree of fit of thecalculated stress gradients to the line is R²=0.9955. It can be seenfrom FIG. 5 that the intrinsic stress gradient approximately increaseslinearity with the hard gold thickness.

Assuming that the residual stress in the soft gold is zero and themodulus of the gold is dominated by the modulus of soft gold (soft goldthickness is 2.3 μm compared to hard gold thickness of between 0.1 and0.3 μm), Stoney's equation can be used to calculate the stress in thehard gold. See G. G. Stoney, “The tension of metallic films deposited byelectrolysis,” Proc., Royal Society, A82, pp. 172-175, 1909.

The effect of aging on the deflection of cantilever beams built inaccordance with the invention was also studied. The beams had a softgold thickness of 2.2 μm and a hard gold thickness of 0.3 μm. The effectof aging was accelerated by subjecting the cantilever beams to annealingtemperatures for extended periods of time. The effect of the annealingtemperature on the cantilever beams was tested by measuring the tipdeflection of the cantilever beams before and after annealing. Thecantilever beams were placed in a vacuum oven at 100° C. for 48 hours (2days). The tip deflection of the cantilever beams was measured using anoptical microscope before and after annealing. The effect of theannealing temperature is given in Table 1, which shows the length, L,width, B, and tip deflection before and after annealing for thecantilever beams studied.

TABLE 1 Average tip deflection, μm L × B Before annealing, Afterannealing   1 mm × 0.5 mm 675 762   2 mm × 0.5 mm 1292 1392 2 mm × 1 mm1333 1438 1 mm × 1 mm 863 945 0.5 mm × 1 mm   367 390

From the average value of the tip deflection for the cantilever beamsbefore and after annealing, it can be concluded that there isapproximately a 10% increase in the deflection of the beams.Recrystallization of the gold in the beams and plastic deformation ofthe gold at the anchor region due to CTE mismatch between the gold andsilicon (CTE of gold=14.7 ppm/° C., CTE of silicon=2.7 ppm/° C.) accountfor the change in tip deflection of the cantilever beams. The error inthe measurement of the tip deflection using this technique was ±5 μm.

The deflection of a cantilever beam built in accordance with the presentinvention was measured as a function of temperature. The cantilever beamhad a soft gold thickness of 2.2 μm, a hard gold thickness of 0.3 μm anda length of 500 μm. The room temperature tip deflection of thecantilever beam was approximately 120 μm. There was negligible change inthe tip deflection when the temperature was lowered to −50° C. and whenthe temperature was raised to 125° C. Therefore, the initial curvatureof the beam was largely independent of temperature in the temperaturerange of −50° C. to 125° C. The error in the measurement was ±5 μm.

In another study, five actuator devices where built in accordance withthe invention using the process in FIG. 3 to study voltage-deflectioncharacteristics of the cantilever beam. The cantilever beams of thesedevices had different shapes and anchor attachments. FIG. 6 shows thetop dimensions (given in μm) of the five cantilever beams, which include(a) a full-hinged square, (b) a full-hinged semicircle, (c) adouble-hinged square, (d) a double-hinged half ellipse, and (e) adouble-hinged rectangular shaped cantilever beam. The darkened regionsin the FIG. 6 devices show the anchor regions for each of the beams. Thearea of the various electrodes in FIG. 6 is the same resulting in asimilar net force applied to each of the electrodes at a given voltage.In each of these actuators, the stationary electrode covered the fullarea under the beam. Each of the five devices in FIG. 6 were tested toevaluate the effect of initial tip deflection, anchor type, and shape ofthe beam on the deflection-voltage behavior of the device. Thedeflection-voltage behavior of the device describes the deflection ofthe beam as a function of the applied bias voltage to the device.

FIG. 7 shows the deflection-voltage behavior of the actuator with thedouble-hinged square beam of FIG. 6(c). The square beam had a soft goldthickness of 2.2 μm and a hard gold thickness of 0.1 μm. The initial tipdeflection of the beam was 206±5 μm (zero applied voltage). At a biasvoltage of 70 V, the square beam was pulled down and made physicalcontact with the bottom dielectric. The solid line in FIG. 7 shows 3-Dmodeling results for the tip deflection obtained using MEMCAD. MEMCAD isa commercial available modeling and design software for MEM devicesdeveloped by Coventor, Inc. The pull-down voltage calculated by the 3-Dmodel was 77.25 V. A stress gradient of 20 MPa/μm over the thickness ofthe beam gave an initial tip deflection of 200 μm using the 3-D model.The curvature produced by the 3-D model was similar to the measuredcurvature of the beam.

FIG. 8 shows the deflection-voltage behavior of the actuator with thefull-hinged semicircular shaped beam of FIG. 6 (b). The beam had a softgold thickness of 2.3 μm and a hard gold thickness of 0.2 μm. Theinitial tip deflection of the beam was 101±5 μm (zero applied voltage).Continuous deflection of the beam occurred up to a bias voltage of 100V. At 100 V, the tip deflection of the beam was 95±5 μm. Upon furtherincrease in voltage to 125 V, the beam was pulled down and made contactwith the bottom dielectric. The center portion of the beam was incontact with the bottom dielectric but the tip was still raised 25±5 μmabove the dielectric. In other words, the center portion of the beamtouched down but not the tip. The solid line in FIG. 8 shows modelingresults for the tip deflection obtained using MEMCAD. The pull-downvoltage as predicted by MEMCAD modeling was 113.25 V.

FIG. 9 shows the deflection-voltage behavior for the actuator with thedouble-hinged elliptical beam of FIG. 6(d). The beam had a soft goldthickness of 2.3 μm and a hard gold thickness of 0.2 μm. The initial tipdeflection of the beam was 245±5 μm. The tip deflection of the beamchanged continuously from 0 V to 70 V, as shown in FIG. 9. When thevoltage was increased to 75 V, the beam was partially pulled down andthe center of the beam made contact with the bottom dielectric resultingin a tip deflection of 150±5 μm. As the bias voltage was furtherincreased to 80 V, the beam further uncurled and had a tip deflection of75±5 μm. The beam was pulled down flat at 85 V. A stress gradient of 40MPa/μm was used to model the beam using MEMCAD. The 3-D MEMCAD modelpredicted that the elliptical beam should remain stable from 0 V to 86 Vfollowed by pull-down at 86.5 V.

FIG. 10 shows the deflection-voltage characteristics of the actuatorwith the full-hinged square shaped beam FIG. 6(a). The beam had a softgold thickness of 2.3 μm and a hard gold thickness of 0.2 μm. Theinitial tip deflection of the beam was 136±5 μm. As the bias voltage wasincreased to 50 V, the tip deflection of the beam decreased to 129±5 μm.The center portion of the beam made contact with the bottom dielectricat 65 V. At 65 V the beam tip was 68±5 μm above the dielectric. Athigher applied voltages, the beam smoothly uncurled to a tip deflectionof 52±5 μm before snapping down. The analysis of this device indicatesthat the initial curvature of the beam deviated from the second orderpolynomial relationship. Once the center portion of the beam madecontact with the bottom dielectric, the remaining length of the beam wasstiffer leading to higher pull-down voltages prior to completesnap-down. The dashed lines in the graph show the 2-D finite differencemodeling of this device. The calculated value of the pull-down voltagewas 62 V compared to the measured value of 65 V. It was observed that at65 V, the remaining movable portion of the beam had an active length of40% of the original length of the actuator. 2-D analysis was performedon the beam with this reduced length, which was assumed to be fixed atone end, so the deflection could occur only at the other end. The radiusof curvature of the reduced length beam was assumed to be the same asthe full-length beam. This gave a calculated tip deflection of 64 μm atthe partial pull-down point compared to the actual value of 68 μm. Thepredicted value of second pull-down voltage (reduced length) was 105 V.

FIG. 11 shows the calculated value of capacitance of this device as afunction of bias voltage. The solid line shows the capacitance vs.applied voltage obtained from the two step pull-down voltage analysis.The initial capacitance of the device was 0.4 pF and increased to 0.53pF at 60 V. Thus, the device had a tunable capacitance range of about32.5% between 0 and 60V. The capacitance value increased to 3.74 pF atfirst snap-down. The capacitance smoothly changed to 3.93 pF at 100 Vfollowed by a maximum capacitance of 5.6 pF.

FIG. 12 shows the deflection-voltage behavior of the actuator with thedouble-hinge rectangular beam of FIG. 6(e). The beam had a soft goldthickness of 2.3 μm and a hard gold thickness of 0.2 μm. As the biasvoltage was increased, it can be seen that the beam snapped down in amanner similar to the full-hinged square beam. The beam had an initialtip deflection of 175±5 μm and was pulled down at a voltage of 40 Vleading to second stable region. The tip deflection of the device at 40V was 105±5 μm. The beam snapped down to its final state at 60 V. A 3-DMEMCAD analysis (dashed lines) gave a pull-down voltage of 35 V. Thelength of the movable portion was of the beam observed to be 70% of thefull beam length. Using this reduced length as the “new” cantileverbeam, the second pull-down voltage was calculated to be 58.5 V comparedto the observed value of 60 V.

The results of the electrostatic analysis of the five actuators aresummarized in Table 2. Table 2 lists the beam shape (square,semicircular, elliptical or rectangular), beam dimensions(length×width), type of anchor attachment (full hinge or partial doublehinge), the initial tip deflection and the pull-down voltage for each ofthe actuator. For the actuators with two step snap-down, only the firstpull-down voltage is presented.

TABLE 2 Tip Dimensions Anchor Deflection Pull-down Shape μm × μm Type μmVoltage Square 857 × 645 Partial 200 70 Semi  547 × 1085 Full 100 125 Circle Elliptical 986 × 762 Partial 250 85 Square 653 × 645 Full 135 60Rectangle 1100 × 471  Partial 175 70

The pull-down voltage of the electrostatic actuators was found to be astrong function of initial curvature, anchor attachment, and shape ofthe beam. The initial deflection and curvature of the beams wascontrolled by varying the built-in stress gradient in the gold layer. Ascalculated by the models, beams of these electrostatic actuatorsexhibited continuous movement with applied voltage before being pulleddown to the bottom dielectric material.

The tip deflection of the beams was dependent on the shape of the beam.The beams were fabricated with curved or straight edges. Due to thebuilt in-stress gradient in the beams, the beams with rectangular andsquare front shapes resulted in corner regions with greater deflectionthan the front edge of the beams. This resulted in higher curvature atthe corners than the center of the beams. The two corners of the beamshad approximately equal deflection. The extra curvature at the cornerswas due to the stress concentration at the corner regions of the beams.The beams with elliptical and semicircular shapes had smooth curvaturesalong their length without any corner effects. These beams showeduniform deflection where the tip of the beams always had the greatestdeflection. The electrostatic actuators with the curved front shapedbeams were found to be more reproducible in shape, which leads to a morereproducible capacitance, and better performance than the square-edgedbeams.

The electrostatic behavior of the actuators with a short (and wide) beamwas compared to the actuators with a long (and narrow) beam (area washeld constant). The pull-down behavior of the double-hinged square beamcan be compared against the double-hinged long rectangular beam (Table2). The dimensions of the double-hinged square beam are 857 μm×645 μmand the dimensions of double-hinged rectangular beam are 1100 μm×471 μm.Both these beams are partially anchored and have approximately the sameinitial tip deflection (200 μm vs. 175 μm). By changing the shape of thebeams from square to rectangular, the pull-down voltage of the topelectrode was reduced from 70 V to 40 V, which corresponds to a 43%reduction in pull-down voltage. Similarly, by comparing the behavior offull-hinge semicircular beam to a full-hinge square beam, it can be seenthat they had comparable tip deflections (100 μm vs. 135 μm) and werefully anchored. A 52% reduction (from 125 V to 60 V) in the pull-downvoltage was achieved by making the beam longer and narrower. Thefull-hinged semicircular beam had greater flexural rigidity than thefull-hinge square beam. The flexural rigidity of the beam is given by(E×I)_(beam), where, E is the biaxial elastic modulus of the materialand I is the moment of inertia. The moment of inertia of the beam isreduced as the beam is made long (and narrow) thereby reducing therigidity of the structure.

The pull-down voltage of the electrostatic actuators was directlyproportional to the initial tip deflection of the beam. The greater thedistance between the beam (top electrode) and the bottom electrode, thegreater the electrostatic force required to move the structure. This canbe shown by comparing the pull-down behavior of the double-hingedelliptical beam and the double-hinged (long) rectangular beam (Table 2).The tip deflection of the long rectangular beam was 175 μm (0.1 μm ofhard gold over 2.3 μm of soft gold). The elliptical beam had an initialtip deflection of 250 μm (0.2 μm of hard gold over 2.3 μm of soft gold).This resulted in an increase in the pull-down voltage from 40 V to 85 V,(an increase of 112%), for corresponding tip deflections of 175 μm and250 μm (a 43% increase). Even though the length of the beam wasshortened from 1100 μm to 968 μm (a 12% decrease), this effect wassmaller than the tip deflection effect.

Partial anchoring of the beam reduced the flexural rigidity of the beam,which resulted in lower pull-down voltages. This can be shown bycomparing the electrostatic behavior of the full-hinge square beam andthe double-hinged square beam. Even though the tip deflection of thedouble-hinged square beam was more than the full-hinged square beam (200μm vs. 135 μm), the pull-down voltage increased only by 10 V (from 60 Vto 70 V). This was due to the partial anchoring of the double-hingedsquare beam. The partial (or double) hinge covered 40% of the width ofthe beam and reduced the rigidity of the structure.

The pull-down behavior of the curved electrostatic actuators was astrong function of the initial curvature of the beams. The secondderivative of the total potential energy with respect to polynomialorder, n is negative for n≦2 indicating unstable behavior. For a furtherdiscussion, see R. Legtenberg et al., “Electrostatic curved electrodeactuators”, Journal of Microelectromechanical Systems, vol. 6, No. 3,September 1997. The actuators with beams created by deposition of 0.1 μmof hard gold showed an initial curvature of polynomial order 2. Thesedevices showed distinct regions of stable and unstable behavior. Thisindicates that the stress gradient model according to the invention isappropriate for beams with small intrinsic stress gradients (20 MPa/μm).

It was observed that as the length of the beam increased, the secondorder polynomial description of the beam curvature became less accurateand deviations from the stress gradient model become more severe. Thebeams with large intrinsic stress gradients fabricated with 0.2 and 0.3μm of hard gold have shown an initial curvature of polynomial ordergreater than 2. The hard gold thickness variation along the length ofthe beam during electroplating resulted in non-uniform stress gradientalong the length leading to the curved beam with higher order curvature.These devices have shown partial snap-down leading to an “uncurling” ofthe beams upon application of voltages greater than the critical pull-involtage. Similar results with step-like unstable and stable regions wereobtained by Legtenberg et al for electrostatic curved electrodeactuators. It has been shown that for beams with curvature of ordergreater than 2, the second derivative of potential energy with respectto the polynomial order becomes positive, indicating stable behavior.The curved beams uncurled along the fixed electrode as the voltage wasincreased, leading to stable behavior (no pull-in) up to the maximum tipdeflection. This behavior of the beams with higher order of initialcurvature is attributed to the constrained beam deflection involvingcontact mechanics. The imperfections at the surface of the beam andbottom electrode also aid in the partial snap-down of the movableelectrode. Surface asperities, entrapped particles or residues betweenthe electrodes after the fabrication process could act like smallbumpers, preventing complete snap-down of the beam.

While various embodiments of the application have been described, itwill be apparent to those of ordinary skill in the art that manyembodiments and implementations are possible that are within the scopeof the present invention. Therefore, the invention is not to berestricted or limited except in accordance with the following claims andtheir equivalents.

All publications cited herein are hereby incorporated by reference intheir entirety for all purposes.

1. A method of fabricating an electrostatic actuator, comprising:forming an electrode on a substrate; forming a support layer over theelectrode; depositing a metal layer on the support layer, comprising thesteps of: depositing a soft metal; and depositing a hard metal onto thedeposited soft metal layer, wherein the deposited soft metal and thedeposited hard metal are made substantially of the same metalcharacterized by different crystal grain sizes and the deposited hardmetal exhibits a higher intrinsic stress than the deposited soft metal.2. method of claim 1, wherein the deposited soft metal and the depositedhard metal are substantially gold.
 3. The method of claim 1, wherein thedeposited soft metal and the deposited hard metal are substantiallysilver.
 4. The method of claim 1, wherein the hard metal is deposited inan electroplating bath containing dopants to enhance the intrinsicstress of the hard material.
 5. The method of claim 1, wherein the hardmetal is deposited at a higher electroplate current density than thesoft metal.
 6. The method of claim 1, wherein the hard metal isdeposited at a lower electroplate temperature than the soft metal. 7.The method of claim 1, wherein the electrode on the substrate iselectrically insulated from the metal layer, permitting the applicationof a voltage between the electrode and the metal layer.